Syntonic comma

Diatonic intervals
Prime second third fourth fifth sixth seventh octave none decime undezime duodecime tredezime semitone / whole tone
Special intervals
Microinterval Comma Diësis Limma Apotome Ditone Tritone Wolf fifth Natural septime
units
Cent Millioctave Octave Savart

The syntonic comma , also didymic comma (after Didymos the musician ), is a small musical interval in music theory , which corresponds to a frequency ratio of 81:80 (approx. 21.51 cents ). The interval plays a role in connection with mood systems and can be obtained by stringing together pure intervals .

Size and frequency ratio

The syntonic comma is the difference between the Pythagorean third (frequency ratio ⁸¹ / ₆₄ ) and the pure third ( ⁵ / ₄ ). A Pythagorean third case corresponds to a ditone two Pythagorean whole tones ( ⁹ / ₈ ):

syntonic comma = Pythagorean third - pure third = 2 Pythagorean whole tones - pure third 21.51 cents . ${\displaystyle \approx }$

Since the frequency ratios are multiplied or divided when adding or subtracting intervals, the frequency ratio of the syntonic comma is calculated as follows:

${\displaystyle \textstyle {\left({\frac {9}{8}}\right)^{2}/{\frac {5}{4}}={\frac {81}{64}}/{\frac {5}{4}}={\frac {81}{80}}=1{,}0125}}$

The syntonic comma also occurs as the difference between a large whole tone ( ⁹ / ₈ ) and a small whole tone ( ¹⁰ / ₉ to):

syntonic comma = large whole tone - small whole tone

Its frequency ratio is calculated as follows:

${\displaystyle \textstyle {{\frac {9}{8}}/{\frac {10}{9}}={\frac {81}{80}}}}$

Syntonic comma as the difference between a third shifted by 2 octaves and 4 fifths

Mid-tone tuning: the syntonic comma is tempered by reducing the fifths

Equal tuning: fifths and thirds are tempered to whole-numbered semitones, which means that the syntonic comma is omitted

Another design option is the difference of perfect fifths 4 ( ³ / ₂ ) to a pure third ( ⁵ / ₄ ) and 2 octaves ( ² / ₁ ):

syntonic comma = 4 perfect fifths - 2 octaves - pure major third

${\displaystyle \textstyle {\left({\frac {3}{2}}\right)^{4}/\left(2^{2}\cdot {\frac {5}{4}}\right)={\frac {81}{80}}}}$

This relationship is the basis of the quarter-point meantone , wherein the fifths to ¹ / ₄ of the syntonic commas are reduced, with the aim that of 4-tempered fifths each a pure third results.

example

The syntonic comma occurs with modulations when the intonation is pure . For example, when modulating from C major to G major, not only  f in  f sharp ( change of sign ), but also a with the frequency 440  Hz increases by a syntonic comma in an  a with 445.5 Hz (frequency ratio ^445.5 ⁄ ₄₄₀ = ⁸¹ ⁄ ₈₀ ).

Modulation from C major to G major

Soprano only

Because this change cannot be made with keyboard instruments , they are medium-tone , well-tempered or - nowadays almost consistently - tuned equally (see Pythagorean comma ).

Web links

• Joachim Mohr: The Pythagorean and syntonic comma